Higher-Dimensional Information Lattice: Quantum State Characterization through Inclusion-Exclusion Local Information

TL;DR

This work extends the 1D information lattice to higher dimensions by defining inclusion-exclusion local information on convex, multi-axis subsystems, assigning information to a position- and scale-lattice so that loop overlaps and redundancies are properly accounted for. The authors implement the framework in 2D with rectangular subsystems and demonstrate its diagnostic power on the 2D Anderson model, a chiral $p_x+ip_y$ superconductor, and the toric code, revealing localized vs. critical scaling, universal edge behavior, and topological entanglement signatures. Across these examples, the approach isolates universal, scale-resolved features of quantum states, including information-based localization lengths, direction-dependent critical exponents, edge-only information scaling, and non-Abelian defect fusion channels. The framework promises broad applicability to mixed states, dynamics, domain walls, and more complex topologies, offering a robust information-theoretic lens on higher-dimensional quantum matter.

Abstract

We generalize the information lattice, originally defined for one-dimensional open-boundary chains, to characterize quantum many-body states in higher-dimensional geometries. In one dimension, the information lattice provides a position- and scale-resolved decomposition of von Neumann information. Its generalization is nontrivial because overlapping subsystems can form loops, allowing multiple regions to encode the same information. This prevents information from being assigned uniquely to any one of them. We address this by introducing a higher-dimensional information lattice in which local information is defined through an inclusion-exclusion principle. The inclusion-exclusion local information is assigned to the lattice vertices, each labeled by subsystem position and scale. We implement this construction explicitly in two dimensions and apply it to a range of many-body ground states with distinct entanglement structures. Within this position- and scale-resolved framework, we extract information-based localization lengths, direction-dependent critical exponents, characteristic edge mode information, long-range information patterns due to topological order, and signatures of non-Abelian fusion channels. Our work establishes a general information-theoretic framework for isolating the universal scale-resolved features of quantum many-body states in higher-dimensional geometries.

Figures (13)

• Figure 1: On a 1D chain of five physical sites [indicated by black arrows at the bottom of (a)], we consider the subsystems $\mathcal{C}_n^\ell$ consisting of $\ell+1$ consecutive sites centered at position $n$ on the chain. Each subsystem is labeled by the indices $(n,\ell)$, which in (a) are represented as square vertices and arranged hierarchically in a triangular structure. Examples are shown for $\mathcal{C}_1^2$ in lilac and $\mathcal{C}_4^0$ in yellow. The local information $i_n^\ell$ is calculated for each vertex $(n,\ell)$ by an addition-subtraction of the subsystem von Neumann information [see Eq. \ref{['eq:local_information']}] as illustrated in (b) for $i_{2.5}^3$. The shaded green area in (b) represents the subsystem $\mathcal{C}_{2.5}^3$. The shaded green area in (a) includes all the vertices associated with subsystems contained within $\mathcal{C}_{2.5}^3$. The total information in $\mathcal{C}_{2.5}^3$ is the sum of all the local information $i_n^\ell$ associated with the vertices within the shaded green area in (a) [see Eq. \ref{['eq:decomposition']}]. The filled green vertices label all the subsystems of scale 1 which are contained in the larger subsystem $\mathcal{C}_{2.5}^3$. The total information at scale $\ell=1$ in $\mathcal{C}_{2.5}^3$ is the sum of all local information $i_n^\ell$ on the filled green vertices [see Eq.\ref{['eq:informationonscaleinaregion']}].
• Figure 2: (a) 1D chain of three qubits denoted $a,b,c$ (bottom) in the state $\rho_{abc}=\tfrac12(|\!\!\uparrow\uparrow\uparrow\rangle\!\langle\uparrow\uparrow\uparrow\!\!| + |\!\!\downarrow\downarrow\downarrow\rangle\!\langle\downarrow\downarrow\downarrow\!\!|)$. The 1D information lattice (top) is represented for this state, where for clarity the names of the subsystems are shown next to the vertices that label them (as $a$, $ab$, and so on). It shows that local information is exclusively located at scale $\ell=1$ in subsystems $ab$ and $bc$ that have 1 bit each, and the total information in the state is 2 bits given by the sum of those two contributions. (b) Consider the same state $\rho_{abc}$ for a system in a triangular arrangement (bottom). There are three two-site subsystem, the new one being $ac$ (green). (Top) On the information lattice based on the inclusion-exclusion information \ref{['eq:inclusion_exclusion']} at scale $\ell=1$ each subsystem contributes 1 bit of information. However, 1 out of these 3 bits is redundant; the 2 bits of information are contained in any pair of $\rho_{ab},\rho_{bc},\rho_{ac}$. Using Eq. \ref{['eq:inclusion_exclusion']}, the overcounting of this redundant information at scale $\ell=1$ is compensated by a negative value at scale $\ell=2$, the scale at which all overlapping subsystems are contained and the dependence in their correlations is resolved.
• Figure 3: (a) On a system of $4\times4$ physical sites (black arrows), each rectangular subsystem is labeled by its scale $\boldsymbol{\ell} = (\ell_x\ \ell_y)$ and the coordinate of its bottom-left corner $\boldsymbol{n} = (n_x\ n_y)$, denoted as $\mathcal{C}_{ {\hbox{$(\mkern-1mu n_x\, n_y\mkern-1mu)$}}}^{ {\hbox{$(\mkern-1mu \ell_x\,\ell_y\mkern-1mu)$}}}$ (for example $\mathcal{C}_{ {\hbox{$(\mkern-1mu 0\,2\mkern-1mu)$}}}^{ {\hbox{$(\mkern-1mu 1\,1\mkern-1mu)$}}}$ in lilac and $\mathcal{C}_{ {\hbox{$(\mkern-1mu 0\,2\mkern-1mu)$}}}^{ {\hbox{$(\mkern-1mu 3\,0\mkern-1mu)$}}}$ in yellow). Each subsystem label is represented by a square marker on a lattice, where they are ordered across scales and positions as shown on the left. (b) For every subsystem, we compute its inclusion-exclusion information $i_{\boldsymbol{n}}^{\boldsymbol{\ell}}$ in Eq. \ref{['eq:rectangular_info']}. The structure of the inclusion-exclusion equation is illustrated for the subsystem $\mathcal{C}_{ {\hbox{$(\mkern-1mu 0\,1\mkern-1mu)$}}}^{ {\hbox{$(\mkern-1mu 3\,2\mkern-1mu)$}}}$. We first take the total information in $\mathcal{C}_{ {\hbox{$(\mkern-1mu 0\,1\mkern-1mu)$}}}^{ {\hbox{$(\mkern-1mu 3\,2\mkern-1mu)$}}}$ and then subtract (blue) the information of the subsystems at scales $(3\ 1)$ and $(2\ 2)$ [first and second lines in \ref{['eq:rectangular_info']}]. The information from all pairwise subtractions (indicated by crossing the green curve) must be re-added (red) [third line in \ref{['eq:rectangular_info']}]. Higher-order intersections are re-subtracted and re-added until only one set is left. The green line separates levels of the inclusion-exclusion order of intersections: the subsystems to the left are intersections of those to the right (marked by the direction of the green arrow).
• Figure 4: (a) We decompose the information of the state \ref{['eq:singlets']}, consisting of all sites in a spin-up configuration except for four sites forming two singlets (illustrated by the blue lines), by evaluating the inclusion-exclusion information in Eq. \ref{['eq:rectangular_info']}. For each vertex $( {\hbox{$(\mkern-1mu n_x\, n_y\mkern-1mu)$}}, {\hbox{$(\mkern-1mu \ell_x\,\ell_y\mkern-1mu)$}})$ (square marker), the inclusion-exclusion information $i_{ {\hbox{$(\mkern-1mu n_x\, n_y\mkern-1mu)$}}}^{ {\hbox{$(\mkern-1mu \ell_x\,\ell_y\mkern-1mu)$}}}$ is represented by a color. The information of the singlets is captured at the smallest scales that fully enclose them (at $( {\hbox{$(\mkern-1mu 0\,0\mkern-1mu)$}}, {\hbox{$(\mkern-1mu 2\,2\mkern-1mu)$}})$ and $( {\hbox{$(\mkern-1mu 0\,3\mkern-1mu)$}}, {\hbox{$(\mkern-1mu 4\,1\mkern-1mu)$}})$ respectively). (b) Summing out $(n_x\ n_y)$ leaves the total scale resolved information of the state $I(\ell_x, \ell_y)$: 32 bits are at $(0\ 0)$ while 2 bits are found at $(2\ 2)$ and 2 others at $(4\ 1)$. (c) Summing out an axis $(n_y\ \ell_y)$ leaves a quasi-1D information lattice over scales $(n_x\ \ell_x)$, where the information of the state is only resolved along the $x$ direction. (d) Summing (b) over $\ell_y$ or (c) over $n_x$ gives the quasi-1D information per scale along $\ell_x$.
• Figure 5: (a) 2D information lattice of the cat state defined in Eq. \ref{['eq:cat_state']}, where the reading of the lattice is identical to Fig. \ref{['fig:singlets']}(a). There are now negative values at scale $\boldsymbol{\ell}=(1\ 1)$, which compensate for the over-counting of information at scales $\boldsymbol{\ell}=(1\ 0)$ and $(0\ 1)$. This signals that the overlapping subsystems at those lower scales have dependent density matrices, what we call an overlap redundancy. (b) The quasi-1D information per scale along the $x$ direction defined in \ref{['eq:quasi-1d-local-scale']} resolves the information along the $x$ direction, and therefore it is insensitive to loop redundancies. Both (a) and (b) show a bit at the largest scale: in cat states a bit of information cannot be inferred by any local measurement and requires access to the entire system.